The kinetic ion-cyclotron instability in the presence of an external DC drift has been investigated. The electron-ion collisions were assumed to be the dominant collisions. The energy-conserving BGK collision operator was used to solve the kinetic equations to obtain expressions for the distribution functions of both species. These expressions, which vary as the wave, were used to obtain the expressions for various plasma parameters. The set of equations thus obtained along with the continuity equations for the two species yielded the general dispersion relation. The dispersion relation was then solved to obtain the expressions for the real as well as the imaginary part (growth rate) of the wave frequency in the various limits of the collisional parameter